1. ## Inverse proportion

Two men, John and Joseph, can paint a house in 12 days, John alone can paint the house in 20 days. How long will it take Joseph to paint the house alone?

2. Did you copy this question properly? At the moment it looks like Joseph doesn't do any work, and so will never get the house painted.

3. oh no, yes i did, thanks for pointing that out, edited

4. Originally Posted by Punch
Two men, John and Joseph, can paint a house in 12 days, John alone can paint the house in 20 days. How long will it take Joseph to paint the house alone?
Two people whose names start with the same letter -- that's evil!

Okay say John's rate is A houses/day and Joseph's is B houses/day.

A + B = 1/12

A = 1/20

Solve for B.

5. $\frac{1}{30}$

6. Originally Posted by Punch
$\frac{1}{30}$
Keep in mind this is not the final answer. Joseph paints 1/30 of a house per day, so in order to paint a full house...

7. 30 days!!

8. Another way, using ratios:

John paints the house in 20 days, so paints $\frac{1}{20}$ of the house every day.

Joseph paints the house in $x$ days, so paints $\frac{1}{x}$ of the house every day.

Together, every day they would paint $\frac{1}{20} + \frac{1}{x}$ of the house every day.

So if we set up a ratio "amount : days"

$\frac{1}{20} + \frac{1}{x} : 1$

$\frac{x + 20}{20x} : 1$

$x + 20 : 20x$

$1 : \frac{20x}{x + 20}$.

So that means they would take $\frac{20x}{x + 20}$ days to paint the entire house. Since we are told that together they take 12 days, that means

$\frac{20x}{x + 20} = 12$

$20x = 12(x + 20)$

$20x = 12x + 240$

$8x = 240$

$x = 30$.

So it will take Joseph 30 days.